How to use Sage to find an explicit isomorphism from a genus 1 curve to an elliptic curve in Weierstrass form?

Question

How can I get Sage to provide an explicit isomorphism from a genus 1 curve to its Weierstrass equation?

I defined the curve to be C = Curve(f, A) where f is a quartic in two variables and A is rational affine space. As C having genus 1 gives an isomorphism from C to Jacobian(C), I then called Jacobian(C) , and Jacobian(C) gives a Weierstrass equation for the curve. I tried reading the Sage documentation for Jacobian, but did not find a way to get an explicit morphism.

I know I need to identify a non-singular point on C to define the group law, and can find such a point algorithmically without much difficulty. Any help is greatly appreciated!

Answer 1

I am not super familiar with the Sage interface, so the following code may be slightly buggy. In the documentation I could not find and analogy to Magma's EllipticCurve command, so instead I will discuss how in general we perform such a computation and give a slightly hacky solution with my little knowledge of Sage.

Consider a genus $1$ curve $C$ over a field $K$. We know that over $\bar{K}$ that $C$ is birational (indeed isomorphic if and only if $C$ is smooth) to its Jacobian $E := \operatorname{Jac}(C)$, which of course can be put in Weierstrass form. I will therefore assume we have extended $K$ sufficiently so that $C(K)$ is nonempty, and fix a nonsingluar point $O \in C(K)$.

Thus $(C, O)$ is an elliptic curve in the sense of Chapter III of Silverman (if $C$ is singular it is not quite, but the fact that $O$ is nonsingular makes it work anyway). The point now is to emulate the proof of Prop III.3.1 of Silverman, to put $(C, O)$ in Weierstrass form.

That is, let $\{1,x\}$ be a basis for the Riemann-Roch space $H^0(C, \mathcal{O}_C(2O))$ and let $y$ be such that $\{1,x, y\}$ be a basis for the Riemann-Roch space $H^0(C, \mathcal{O}_C(3O))$. The same arguemnt as Prop III.3.1 shows that $x,y$ must satisfy a Weierstrass equation.

So we just have to force Sage to compute the Riemann-Roch basis. Now while writing this I noticed that I can't find a general implimentation for this on Schemes in Sage (one does exist in Magma) and perhaps this is why such a function does not exist? I would not know. In any case, for curves over finite fields such a function does exist. I'll give an example now where I put $C: y^2 = x^4 - 1$ over $\mathbb{F}_5$ into Weierstrass form with the point $O = (1,0)$:

sage: P2.<U,V,W> = ProjectiveSpace(GF(5),2)
sage: C = Curve(W^2*V^2 - U^4 + W^4)
sage:
sage: D = C.divisor(C(1,0,1))
sage:
sage: C.riemann_roch_basis(2*D)
[1, (U^3 + U^2*W + V^2*W + U*W^2 + W^3)/(U*V^2)]
sage: x = C.riemann_roch_basis(2*D)[1]
sage:
sage: C.riemann_roch_basis(3*D)
[1,
 (U^4 - U^2*V^2 + U^3*W - 2*U*V^2*W + U^2*W^2 + V^2*W^2 + U*W^3)/(V^3*W),
 (U^3 + U^2*W + U*W^2 + W^3)/(V^2*W)]
sage: y = C.riemann_roch_basis(3*D)[1]
sage:
sage: h = Hom(C,P2)
sage: f = h([x,y,1])
sage: f.image()
Closed subscheme of Projective Space of dimension 2 over Finite Field of size 5 defined by:
  U^3 - U^2*W + V^2*W + U*W^2 - W^3
sage: